Microwave Imaging for Conducting Scatterers by Hybrid Particle Swarm Optimization with Simulated Annealing

2011 8th International Multi-Conference on Systems, Signals & Devices

MICROWAVE IMAGING FOR CONDUCTING SCATTERERS BY HYBRID PARTICLE SWARM OPTIMIZATION WITH SIMULATED ANNEALING

Bouzid Mhamdi, Khaled Grayaa and Taoufik Aguili

Syscom Laboratory, Engineer School of Tunis - BP 37 Belvedere, Tunis - 1002, Tunisia [email protected]

ABSTRACT The first is based on gradient searching schemes such as the Newton-Kantorovitch method [3], modified In this paper, a microwave imaging technique for gradient method [4] and Levenberg-Marguart algorithm reconstructing the shape of two-dimensional perfectly [5]. It is well-known that these deterministic techniques conducting scatterers by means of a stochastic used for fast reconstruction of microwave images tend to optimization approach is investigated. get trapped in a local extreme and suffer from a major Based on the boundary condition and the measured drawback, where the final image is highly dependent on scattered field derived by transverse magnetic the initial trial solution. To overcome this obstacle, a illuminations, a set of nonlinear integral equations is second class of population based stochastic methods such obtained and the imaging problem is reformulated in to as the genetic algorithm (GA) [1-2], simulated annealing an optimization problem. (SA) [6] and neural network (NN) [7] has become A hybrid approximation algorithm, called PSO-SA, is attractive alternatives to reconstruct microwave images. developed in this work to solve the scattering inverse These techniques consider the imaging problem as a problem. In the hybrid algorithm, particle swarm optimization (PSO) combines global search and local global optimization problem, and reconstruct the correct search for finding the optimal results assignment with image by searching for the optimal solution through the reasonable time and simulated annealing (SA) uses use of either rivalry or cooperation strategies in the certain probability to avoid being trapped in a local problem space. optimum. Recently, the particle swarm optimization (PSO) The hybrid approach elegantly combines the exploration algorithm [8] has attracted the interest of the ability of PSO with the exploitation ability of SA. electromagnetic community as a promising technique for Reconstruction results are compared with exact shapes of the solution of optimization and design electromagnetic some conducting cylinders; and good agreements with the problems [9]. Actually, the PSO algorithm has already original shapes are observed. been applied to the solution of inverse scattering problems related to the reconstruction of dielectric Index Terms— Hybrid algorithms, Microwave scatterers [10-11]. imaging, Particle Swarm Optimization, Simulated It has been reported that PSO has better performance Annealing, Shape reconstruction in solving some optimization problems. However, basic PSO algorithm suffers a serious problem that all particles are prone to be trapped into the local minimum in the later phase of convergence. Although the SA Algorithm 1. INTRODUCTION has been proved that it could convergent to overall optimal solution by using probability, but more time is The application of electromagnetic scattering to retrieve taken in seeking high-quality approximately optimal the shape, location, size, and the internal property of an solution [12]. object embedded in a homogeneous space or buried In this paper, we proposed a new hybrid optimization underground has gained increasing interest in many areas approach which combines PSO algorithm with the SA such as non-destructive evaluation, geophysics, algorithm and apply it to solve the scattering inverse biomedical applications, material engineering, remote problem. By integrating SA to the PSO, the new sensing, and environmental investigations. A particular algorithm, which we call it PSO-SA makes full use of the problem of this kind is the estimation of the location as strong global search ability of PSO and the strong local well as the shape of conducting scatterers [1-2]. search ability of SA and offsets the weaknesses of each However, it is well known that the main difficulties other. of inverse problems are the nonlinearity and ill-posed. As The PSO-SA approach is applied to the a result, many inverse problems are reformulated as reconstruction of the shape of perfectly conducting optimization problems. Much attention has been paid to scatterers. The principal objective of the proposed developing the inversion algorithms and a variety of technique is to minimize a cost function that describes the algorithms have been proposed. Generally speaking, two discrepancy between the measured and estimated main kinds of approaches have been developed. scattered field data. 978-1-4577-0411-6/11/$26.00 ©2011 IEEE Measurement Incident 2. PROBLEM FORMULATION

2.1. Forward problem

The geometrical configuration is depicted in figure 1. We consider a Perfectly Electrically Conducting (PEC) cylinder with circular cross section placed in a homogeneous background medium. The cylinder is assumed infinite long in z direction, while the cross-section of the cylinder is arbitrary, so a 2D approximation can be used. The scatterer is illuminated by a circumference source of pulsed plane waves, with z-polarized incident electric field El (TM polarization) and a propagation direction impinges at an incident angle p l in the (x, y) plane. The z polarized incident wave can be given by (1): Figure 1. Geometry of the considered inverse scattering El(x,y) — g-jKo&cosvi+y.sinvi)' ( 1) problem in (x, y) plane. Where where Ko — w ju o- eo denotes the free-space wave number. 8 — / p 2(p) +~p2(pTf—'2pXpJpXp') cos(p—~pT) (8) In this work, the shape of the object can be expressed by means of a Fourier series of order Q as follow: f — J p 2(y') + p'2(p") (9)

Conventionally, (7) is solved by employing the p (p ) — 1 an cos(np) + 1 bn sin(n^) (2) method of moments (MoM) [13] in which the contour of the cylinder is divided into R segments. The current n = 0 n = l density is assumed to be constant in each segment and a an and bn represent the Fourier coefficients: matrix equation is derived to solve for the unknown current coefficients. One should not forget that the direct 1 f 2n scattering problem is to calculate the scattered electric an — — I p(p) cos(np) dp (3) fields while the shape and location of the scatterer are n -*0 given. Then, the scattered field is given as:

1 f 2n IX bn — I p (p ) sin(n^) dp (4) MUo V "1 n Jo Es(r,p) — ---- 4 ~ A p 1 jn H (<0l)(Ko8n)fn (10)

The total and scattered electric fields are polarized along the z-axis, thus the problem is reduced to a TM Where Ap is the angular discretization size, 8n and fn are scalar formulation. Starting from Maxwell’s equations, a given as follow: Freedom integral equation of first kind may be derived in which the scattered field EsSz is given as: 8n — jr 2 + p2(pn) — 2rp(pn) COS(p — pn) (11)

Es(r) — - ^ 0 j Jz(r')H0(Ko\r — f'|)df' (5) fn — j p2(pn) + p'2(pn) (12)

2.2. Scattering inverse problem Where r and r denote the observation and field points, Jz is the induced current density parallel to the z-axis, and Let us assume that the scatterer is illuminated by a H1 is the Hankel function of the first kind and zeroth number of incident fields and that for each incidence the order. At the surface of the cylinder, the total electric scattered field is measured at a set of measurement field satisfies the boundary condition: positions around the scatterer domain. The objective of the microwave imaging, i.e. the inverse problem is to E l + E S — 0 (6) estimate the scatterer contour from the total set of measurements. To cope with this inverse scattering Considering that r — (p(p),p) on counter C, if we problem, we define a cost function representing the combine (5) and (6), one obtains: discrepancy between measured and estimated scattered field. In this study, the inversion procedure is based on MUo f , „ El(p(p'),p') —4° | Jz(p)Ht(Ko8)fdp (7) minimization of the cost function F which represents a relative error with respect to the Fourier coefficients. In each iteration, the velocity and position of each S meas s 2 1 V V \^im ^im\ F(X)_ (13) particle are updated according to its best encountered meas \ position and the best position encountered by any \E

X _ [a0, ai,..., aQ, bv b2 ,...,bQ] Vi.d _ w. Vu + Ciri(Pi,d - Xiid) + C2rz{Pg,d - Xi,d)(14)

Where S is the total number of incidences, M is the Xi,d _ Xi,d + Vi,d (15) number of measurements per incidence, Emeas denotes the measured scattered field and Es is the estimated one. where w is the inertia weight. cl and c2 are the Actually, since the contour is a function of X, (13) is acceleration coefficients and the parameters rl and r2 are minimized with respect to X. two random numbers distributed uniformly in [0,1]. In this work, this problem is resolved by an Generally, the value of Vid is restricted in the interval optimization approach, for which the global searching [-Vmax,Vmax], Vmax is decided by the user. scheme PSO and its hybrid model with the SA heuristic The inertia w is used to achieve a balance in the method are employed to minimize the above cost exploration and exploitation of the search space. The function (F). inertia dynamically reduces during a run which facilitates a balance in the exploration and exploitation of the search 3. RELATED ALGORITHMS space.

3.1. Particle Swarm Optimization 3.2. Simulated annealing

Particle swarm optimization (PSO) is a population based stochastic optimization technique developed by Eberhart Simulated annealing (SA) was proposed by S. Kirkpatrik, and Kennedy in 1995 [8], inspired by social behavior C. D. Gelatt and M. P. Vecchi in 1983 [12]. It was one of patterns of organisms that live and interact within large the first minimization stochastic methods applied to groups. In particular, it incorporates swarming behaviors electromagnetic imaging. SA is a probabilistic variant of observed in flocks of birds, schools of fish, or swarms of the local search method, but it can, in contrast, escape bees, and even human social behavior. local optima. SA is based on an analogy taken from The traditional PSO model consists of a number of thermodynamics: to grow a crystal, we start by heating a particles moving around in the search space, each particle row of materials to a molten state. Then, we reduce the representing a possible solution to a numerical problem. temperature of this crystal melt gradually, until the crystal The position of a particle is influenced by the best structure is frozen in. A standard SA procedure begins by position visited by itself i.e. its own experience and the generating an initial solution at random. At initial stages, position of the best particle in its neighborhood. When a small random change is made in the current solution Xc. the neighborhood of a particle is the entire swarm, the Then the objective function (representing the cost best position in the neighborhood is referred to as the functional F) value of the new solution Xn is calculated global best particle, and the resulting algorithm is referred and compared with that of the current solution. to as the gbest PSO. When smaller neighborhoods are A move is made to the new solution Xn if it has used, the algorithm is generally referred to as the Ibest better value or if the probability function implemented in PSO. The performance of each particle is measured using SA has a higher value than a randomly generated number. a fitness function that varies depending on the Otherwise a new solution is generated and evaluated. The optimization problem. In a space of D dimensions, each probability p of accepting a new solution which called particle in the swarm is represented by the following Metropolis law is given as follows: characteristics: Xi = [Xi1, Xi2,....,XiD]: The current position of the particle; Vi=[Vi1, Vi2,..,V iD]: The current 1 ifF(Xn)-F(Xc)<0 velocity of the particle; P; = [pi1, pi2,..., piD]: The personal p _ -\F(Xn)-F (X c)\ (16) best position of the particle. The personal best position of exp otherwise particle i is the best position visited by particle i so far. There are two versions for keeping the neighbors’ best T is a control parameter (temperature), which is vector, namely Ibest and gbest. In the local version, each updated during the iterative process by a fixed scheduling particle keeps track of the best vector Ibest attained by its rule (usually, of logarithmic type). The calculation of this local topological neighborhood of particles. For the probability relies on a temperature parameter T, which is global version, the best vector gbest is determined by all referred to as temperature, since it plays a similar role as particles in the entire swarm. Therefore, the particles the temperature in the physical annealing process. To have the tendency to move towards the better and better avoid getting trapped at a local minimum point, the rate search area over the course of search process. PSO of reduction should be slow. In our problem the following algorithm starts with a group of N random (or not) method to reduce the temperature has been used: particles (solutions) and then searches for optima by updating each generation. T(n + 1) _ p.T(n) (1 7) where the annealing rate satisfies 0 < ft < 1 . between 0 and 1), the new shape contour is also accepted, In this paper, the initial temperature is determined by otherwise the new shape will be rejected. the following empirical formula: Combining the fast optimal search ability of PSO with the probability jump property of SA, we design a Fmax Fmln new algorithm frame to solve the shape reconstruction To — - (18) ln(0.1) problem for PEC objects. The main idea is that at first every particle Xl searches its local best contour where Fmax and Fmin are the maximum and minimum Xllbestusing SA algorithm to update individual personal objective values of the solutions in the initial swarm, best contour Pl and the global Pg. respectively. The steps of PSO-SA are given below: Thus, at the start of SA most worsening moves may Stepl: Initialization. be accepted, but at the end only improving ones are likely Initialize the particles number N, Vt and Xt of each to be allowed. This can help the procedure jump out of a particle, the upper and lower values of the inertial factor local minimum. Simulated annealing is in principle able W. Wmaw and Wmln. to reach the global minimum, but is computationally very Configure T0, Tf, ft and LT. T0 is the initial temperature heavy, like most of the stochastic optimization given by (18). Tf is the lowest temperature value. At low procedures. temperature, every particle finds its local best contour in its local area. ft is cooling coefficient which is a random 4. PSO-SA HYBRID ALGORITHM constant between 0 and 1. LT is maximum number of iterations in a certain temperature. To overcome trapping into local minimum problem and Step2: Every particle Xl searches for its local best increase the diversity of particle swarm, the simulated contour Xabest annealing algorithm is introduced into PSO to solve the In one iteration, every particle Xl generates a new contour scattering inverse problem. Xl in its local area and then according to the accepting In PSO algorithm, particles always chase the current rule of SA decides whether to accept the new contour or overall optimal point and history optimal point found not. After LT iterations, every particle finds its local best heretofore [14]. Then the particle speed closes to 0 contour Xabest. quickly and can not escape from local minimum. In order Step3: Update personal best contour Pl and the global to avoid earliness convergence, the algorithm must best contour Pg. escape from local minimum and search in other solution For each particle, the adaptive fitness value F(Xllbest) is space, until solve overall optimal solution. Simulated compared with one of the historical best position Pl, if the annealing algorithm accepts a new value by certain adaptive value is better than one of Pl. Then, Xllbest is probability, accept a worse, it has the ability of escaping consider as the best position Pl, otherwise, Pl remain from local optimal solution and can restrain earliness unchanged. convergence, increase the diversity of the PSO. This hybrid approach makes full use of the llbest F(Xubest) < F(Pl) exploration ability of PSO and the exploitation ability of (19) SA and offsets the weaknesses of each other. F(Xlibest) > F(Pl) Consequently, through introducing SA to PSO, the After updating every particle’s personal best value, we proposed algorithm is capable of escaping from a local can get the new global best value Pg. optimum. And particle swarm optimization-simulated annealing (PSO-SA) hybrid algorithm become a global Step4: The position and speed of each particle were optimal approach by using this new acceptance rule, the updated following the function (14) and (15). theory has been proved. Step5: Calculate the new temperature T specified in The basic idea of PSO-SA algorithm is shown below: equation (17). If T < Tf. the algorithm finds the best Starting from an initial solution, the algorithm contour Pg and comes to the end. Otherwise go to Step2. unconditionally accepts any solution which results in smaller fitness function value than the last solution. In 5. NUMERICAL RESULTS this paper, the decision to accept or reject a new solution with larger fitness function value is made according to the probability function given by (16). In this section, we discuss reconstruction results using Equation (16) selects solutions according the value synthetic data. We illustrate the performance of the AF — F(Xn) — F(XC). When AF is non-positive, which proposed inversion hybrid algorithm PSO-SA and its means that the Fourier coefficients sensitivity to random noise in the scattered field. The [ao, ai,... , aQ ,bl,b2, ...,bo\ values of new contour are measurement is simulated by computation of the scattered closer to those of real contour than the old one, the new field with preset electromagnetic parameters, by means of object shape, i.e, the Fourier coefficients, is always the method of moments (MoM). The scatterer is illuminated by 5 TM plane wave incidences (S=5), accepted. In the case of AF > 0, firstly the acceptable uniformly distributed. The frequency is chosen to be probability should be calculated according to equation 1GHz, the wavelength is Ao — 0.3m. (16). If P > (rand (0,1) is a number generated randomly For each incidence, the scattered field is measured at ■ Fteccnstructicn by RSO M—20 positions, uniformly placed around the scatterer at - Rscocetnicticn by RSOSA a distance from the scatterer center equal to 6A0 . ■ Exaot shape The object which one will reconstruct its unknown 90 O.OB shape is divided into 36 segments (R—36). The shape of the cylinder is expressed with Fourier coefficients (Q—5), thus the size of a particle is equal to 11—2Q+1. The initial values of the Fourier coefficients were chosen randomly (uniformly distributed in the range [-1, 1]). The initial values of the components of the particle velocities were also chosen randomly (uniformly distributed in the range [-0.2, 0.2]). The population size N of the swarm was selected equal to 60. The acceleration coefficients were selected constant (c1—c2—0.8), whereas the inertia weight w decrease linearly starting from 1 to 0.5. For SA algorithm, we used the following configuration: initial temperature T0 is given by (18), the lowest temperature 7y —0.0001, cooling coefficient ft —0.9. To show the effect of noise quantitatively, a relative error of shape function between the reconstructed profile and the true one is defined as Figure 2. Shape reconstruction results for the first example.

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F- ] i \ [ p Cal(9r) ~ p trUe(9r)]2/ (20) t ~)RL / P2(

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